Wavelet-Based Distributed-Order Optimal Control Problem

Abstract

This research presents a novel numerical method for solving a class of complex optimal control problems that feature distributed-order derivatives. The approach leverages the advanced mathematical properties of fractional-order Laguerre wavelets to transform the original continuous-time control problem into a more manageable discrete set of algebraic equations. This transformation is accomplished by utilizing Reimann-Liouville distributed-order operational matrices, which serve as the foundation for accurately representing the system's dynamics. In addition, the method incorporates a strategically chosen set of Newton-Cotes collocation points, which enhances the precision of the discretization process and ensures that the system's behavior is faithfully captured. Once the system is discretized, the optimized solution is determined by applying the Lagrange multiplier method. This powerful optimization technique is used to efficiently solve the resulting system of equations, ensuring that the solution adheres to the necessary constraints. The method not only facilitates the conversion of the original continuous-time problem into a discrete framework but also guarantees that the optimality conditions are systematically satisfied throughout the process. Finally, the effectiveness and versatility of the proposed numerical method are demonstrated through a series of illustrative examples. These examples are carefully selected to showcase the method's high precision, robustness, and broad applicability to a wide range of generalized distributed-order optimal control problems. The results highlight the method's ability to handle various types of distributed-order dynamics, confirming its potential for tackling challenging problems in modern control theory and engineering practice. Overall, this study provides a significant contribution to the field by presenting a systematic and practical approach for solving distributed-order optimal control problems. The combination of fractional-order Laguerre wavelets, operational matrices, and advanced optimization techniques results in a highly effective computational tool that can be readily adapted for a variety of applications. This approach is particularly effective for multi-echelon supply chains, where coordinated control strategies across suppliers, manufacturers, and distributors lead to improved stability, reduced bullwhip effects, and enhanced overall performance of the supply network.